In Days of Future Past, Quicksilver holds Magneto's head to prevent whiplash. However, in Apocalypse, Quicksilver not only ignores holding heads (not everyone), he also throws people. Did they make a movie mistake, or is there some explanation of this?

Not sure what exactly the situation in Days of Future Past was, but in Apocalypse he had a lot of people to save in a relatively short amount of time - probably not enough to be able to worry about that for everybody he was saving. Alternatively, it wasn't actually required in DoFP, he just didn't know that yet so was playing it safe. This question suggests that in reality holding the head is probably irrelevant and they're still going to suffer fatal damage. Ultimately, it's comic book physics plus the Rule of Cool.
– Anthony GristOct 17 '16 at 15:23

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Because the movies are not self consistent. They are not self consistent because they are not written to be self consistent. They are not written that way because the studio values the instantaneous perception of "hey that is cool" over their concerns that we will find plot holes later. They value that because that allows for easy trailers and they believe it is better for casual fans. The net result is they do not develop as strong a fan base as MCU films but make comparable profits. This is all out of universe. In universe: Grist's genius 2nd idea or he finds time to decelerate in Apoc.
– kaineOct 17 '16 at 21:35

2 Answers
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In Days of Future Past, Quicksilver was trying to get one important guy out of jail. Key aims: keep Magneto safe, and get him out in the few minutes before the whole place is on lockdown.

In Apocalypse, Quicksilver was trying to get every student out of an already-exploding school. Key aims: get everyone away from the explosion in the one or two seconds before it engulfs and certainly kills them.

As such, he doesn’t have time to hold everyone’s head. More risk of injury for any given individual is acceptable, given that the alternative is immediate firey death, and he’s got a lot of kids he wants to save in a very short period of time. As you'll notice, some kids he just literally throws out onto the ground.

I can't remember him doing that in the film. But it may be, as someone has already suggested, that he was playing it safe. It may be important to note that Magneto is Quicksilvers father. This is alluded to in DoFP and confirmed in Apocalypse. This may be why he was more cautious with Magneto than he was with others.

As far as I know, Quicksilver did not know Magneto is his father in DoFP. He might suspect but did not know for certain. The dialogue in the elevator goes something like this: "You can control metal? My mother knew a guy once who also could control metal."
– Scorch91Oct 18 '16 at 10:25

It was never confirmed whether he did or did not know at that point. Nonetheless, the allusion exists to poke fun at the fact that Magneto was his father in some comics and to reference those comics. The fact that he acknowledges in Apocalypse that Magneto is his father suggests to me that it was a little more than an inkling that he had. This explains why he would be more careful with Magneto than he was with others.
– TJF.Oct 18 '16 at 10:35